Abstract

The exact boundary controllability of a class of enhanced oil recovery systems is discussed
in this paper. With a simple transformation, the enhanced oil recovery model is first affirmed to be neither
genuinely nonlinear nor linearly degenerate. It is then shown that the enhanced oil recovery system with
nonlinear boundary conditions is exactly boundary controllable by applying a constructed method. Moreover, an interval of the control time is presented to not only give the optimal control time but also show the time for avoiding the blowup of the controllable solution. Finally, an example is given to illustrate the effectiveness of the proposed criterion.

1. Introduction

In recent years, the economy has developed rapidly over the years requiring a lot of energy sources in China, but it is impossible to largely import oil required. Many oil fields in China are developed by water flooding, but now, the recovery efficiency is low and water cut is over because of the heterogeneity of reservoirs and high viscosity of oil. It is essential to increase the oil production of oil fields. As a result, enhanced oil recovery (EOR) has been a challenging field for different scientific disciplines. A mathematical model in [1] is developed to describe surfactant-enhanced solubilization of nonaqueous-phase liquids (NAPLs) in porous media. The goal in [2] is to find an optimal viscosity profile of the intermediate layer that almost eliminates the growth of the interfacial disturbances induced by mild perturbation of the permeability field. The mechanism of enhanced oil recovery using lipophobic and hydrophilic polysilicon (LHP) nanoparticles ranging in size from 10 to 500 nm for changing the wettability of porous media is analyzed theoretically in [3]. It is shown in [4] that water-soluble hydrophobically associating polymers are reviewed with particular emphasis on their application in improved oil recovery (IOR). The solution properties of enhanced oil recovery are provided in [5, 6]. In order to enhance oil recovery and stabilize oil production, the study on EOR has been carried out for more than 20 years (also see [7–11]).

One of the strategies used in EOR is to use polymer flooding. Polymer flooding involves using a polymer additive to increase water viscosity, improve the water-oil mobility ratio, and enhance the displacement efficiency. Polymer flooding has been widely applied as an effective tertiary oil-recovery method in Daqing, Shengli, and other oilfields in China.

However, Different polymer flooding units have different static conditions and development status before polymer flooding. The production performance and behavior are also different. The quantitative characterization and prediction of polymer flooding performance have important guiding significance for polymer flooding scheme programming, performance evaluation, and adjustment. Hence, it is necessary to construct some mathematical models to illustrate the properties of polymer flooding. In [12–15], a nonlinear system model is presented to describe the polymer flooding of an oil recovery:
where is the saturation of the aqueous phase (i.e., the solution of polymer and water, 1), is the concentration of polymer in the water (). [16] is the particle velocity of the aqueous phase. denotes the position in the reservoir and denotes the time. In the polymer flooding, water thickened with polymer is injected into the reservoir.

Let ; system (1) can be written as
where , is a scalar function and usually referred to be the flow function. In this paper, we consider to be rotationally invariant; namely, define that with . As a result, system (2) can be described as

To the authors’ knowledge, seldom researchers discussed the optimal control problem of system (3) (or (1)), but it is really interesting. Actually, in the last forty years, different optimal control schemes such as pinning control and impulsive control have been presented on all kinds of mathematical models of the engineering and physical application [17–19]. It is worth noting that almost all of the discussed models in [17–19] are ordinary differential but system (3) is partial differential. A problem is arisen: how to discuss the control problem of the partial differential model (3)? By the constructive method, the authors in [20, 21] discuss the global exact boundary controllability of a class of quasilinear hyperbolic systems of conservation laws with linearly degenerate characteristics. Inspired by [20, 21], we will discuss the exact boundary control problem of system (3) by using a constructed method. Hence, the main concern of this paper is to design an exact boundary controller for the EOR model (3).

The remainder of this paper is organized as follows. In Section 2, the exact boundary control problem and some Lemmas are presented. In Section 3, the main result is completed by a constructive method. Moreover, some important lemmas are also proposed in this section. In Section 4, an example is carried out to illustrate the effectiveness of the main result. Finally, conclusions are drawn in Section 5.

2. Problem Description

Let ( and is a unit vector); system (3) can be written as
one has
According to (5), one has
Then, one has from (6) and (7)
Inserting (8) into (6) or (7), one gets
Hence, one has
where .

In the following, we will investigate the exact boundary control problem for system (10) (or system (3)). Consider system (10) posed on the domain
with the initial data
and the nonlinear boundary conditions
where are given smooth functions. systems (10) and (13) can be viewed as boundary control systems when boundary value functions and are considered as control inputs. Hence, we only need to study the following problem.

Exact Boundary Control Problem: given , , and , , can one find a time and control inputs , such that the boundary control systems (10) and (13) have a solution satisfying the initial conditions (12) and the terminal conditions
In order to solve the above boundary control problem, we need the following assumptions and Lemmas.

Assumption 1. For any , there exists

Assumption 2. For simplification, we assume that
As a result, when , the Cauchy problem (10) and (12) has a global solution. Moreover, .

Remark 4. The discussed models in [20, 21] are both linearly degenerate. However, model (10) in this paper is neither genuinely nonlinear nor linearly degenerate, which is more difficult and complicated to be discussed.

Lemma 5 (see [22]). Consider the Cauchy problem (10) and (12). Suppose that , , , are all functions and the norm of and are bounded. Under Assumption 1, if
the Cauchy problem (10) and (12) has a unique global solution on the domain .

Remark 6. For the Cauchy problem (10) and (14), we need to modify (17) to be
When , the Cauchy problem (10) and (14) has a unique global solution on the domain . Moreover, .

In the following, we consider the Goursat problem of system (10) on the angular domain
We prescribe boundary conditions
where and are the characteristics passing through the origin point , on which it holds

Lemma 7 (see [22]). Suppose that , , , are all functions. Under Assumptions 1 and 2, if
the Goursat problem (10) and (20) has a unique global solution on the domain .

First, we need to discuss the lifespan of the Cauchy problem and Goursat problem. From the Cauchy problem (10) and (12), we have the following lemma.

Lemma 8. If there exists such that , the Cauchy problem (10) and (12) must blow up in a finite time and the lifespan is dependent on the initial data.

Proof. For the first equation of system (10), the characteristic can be defined by
One has ; that is, . That is, the norm of is finite. Hence, we need to show that the first derivative of must blow up in a finite time.Clearly, . From (23), one has . Then,
According to (16), if there exists such that , one has
That is, the Cauchy problem (10) and (12) must blow up in a finite time. Moreover, from (24), one has when , which means that the lifespan is dependent on the initial data. The proof is completed.

Remark 9. Note that, in the second equation of system (10), according to [22], will always be bounded. Moreover, will blow up if the following holds:
Obviously, this does not hold since is independent of the function . As a result, will never blow up in a finite time.

Lemma 10. If there exists such that , the solution of the Goursat problem (10) and (20) must blow up in a finite time and the lifespan depends on the initial data.

Proof. For , its two characteristics have two intersect points with the curves and , which are, respectively, defined as and . Along the two characteristics, one has and , respectively. As a result, are bounded.In the following, we will calculate and . Clearly, one has , and . Hence, . According to (15) and (16), one has the fact that and . As a result, if there exists such that , in a finite time . Moreover, similar to Lemma 8, one knows that the lifespan depends on the initial data.Similarly, one has with . As a result, = , which means that will never blow up in a finite time.

Remark 11. According to Lemmas 8 and 10 and Remark 9, one knows that the blowup of the Goursat problem and the Cauchy problem only occurs in the solution .

3. Main Results

In this section, the boundary controllers will be designed.

Theorem 12. With Assumptions 1 and 2, and conditions (17)-(18), for given , , , in the space with their norms bounded by and the small norms of their first derivatives, and for any satisfying , there exist such that systems (10) and (12) have a solution on the domain
satisfying , , , , for , where
and . Moreover, is the lifespan of the Cauchy problem (10) with the initial data on .

Proof. One has the following.Step 1. Discuss . Let , , , (see Figure 1). Let be the intersection point of the lines
Let be the intersection point of the lines
Let the curves and be, respectively, described by the characteristics and (see Figure 1), which satisfy
Define
Let be the intersection point of the curves and . Similarly, let be the intersection point of the curves and . Here, curves and are, respectively, described by the characteristics and (see Figure 1), which satisfy
Define
Here, we have to satisfy two conditions.(1)The lines and have no intersection point in .(2)The lines and have no intersection point in .Otherwise, if condition (1) is not satisfied, there exists a characteristic which passes through two points and , . According to the characteristic property, one has , for and is enclosed by the characteristics , , and the axis.Note that the initial and terminal conditions are usually to be arbitrarily chosen. If one choose that , the system will not go from the given initial state to the desired terminal state no matter what control inputs are given. As a result, condition (1) should be satisfied. Also, with the similar analysis, condition (2) should be satisfied.Let be the intersection point of the lines and ; one has . From condition (1), one has the fact that . Let be the intersection point of the lines and ; one has . From condition (2), one has that . As a result, . Hence, and .Remark 13. (i) In [20, 21], the authors require that . Actually, we only need that = . Here, = , , and = − .If , then , and . One has the fact that
From the above inequality, one has . Similarly, if , one can obtain that .Hence, can be written as . (ii) denotes that the solution of the Cauchy problem does not blow up in the domain .Step 2. Discuss the Cauchy problem in the domains and . Let the domain be enclosed by the characteristics and and the axis (see Figure 1). Let the domain be enclosed by the characteristics and and the horizontal line (see Figure 1). From Lemma 5 and Remark 6, the Cauchy problem (10) and (12) (or (10) and (14)) has a unique global solution on the domain (or ). Moreover, one has(1)on the straight line , , and on the curve , ;(2)on the straight line , , and on the curve , .Step 3. Let be the domain enclosed by the characteristic , the characteristic , the straight line , and the straight line , where is denoted by
where is the slope of the straight line . Consider the following system on the domain :
with initial conditions
For points and , it is required that
Along the characteristic , one has
Along the straight line , one has
So, it is required that
Similarly, along the characteristic , one has
Along the straight line , one has
It is therefore required that
In addition, along the characteristic , one has
Along the straight line , one has
So, it is required that
Similarly, along the characteristic , one has
Along the straight line , one has
It is therefore required that
Moreover, the following compatibility conditions of points , , , are also required:
According to Proposition 2.1 of [21], one has the fact that
when . Then, from Assumption 2 and Remark 6, one has , . With Lemma 8, the Cauchy problem (37) and (38) with prescribed data on must blow up in a finite time. Here, we have interchanged the role of and variables. Hence, the time means the -axis.

In what follows we will prove that we can choose an appropriate vector to satisfy that the first blowup point is out of .

Lemma 14. One can choose such that the first blowup point satisfying ; that is, the Cauchy problem (37) and (38) has solution on the domain .

Proof. , two characteristics passing by can be defined by
where and are the intersection points of the two characteristics and the straight line . Along the characteristic , one has the fact that . As a result,
From (54), one has . According to Lemma 10 and Remark 9, one knows that never blow up. Hence, we only need to estimate . From (56), one has and = − . From (53), one can find such that . Note that in , if . So, we only need to discuss .When , there exists such that . As a result, if we choose satisfy that
then, the blowup point satisfies that and .

Remark 15. For convenience, conditions (17) and (18) are chosen to guarantee that there have been no blowup points in and . Actually, without conditions (17) and (18), we can also get the solution in the domains and if we use the above method to obtain a similar condition of (57).

Similarly, we have the following lemma for the domain .

Lemma 16. One can choose such that the first blowup point satisfying , that is, the Cauchy problem (37) and (38), has solution on the domain .

Proof. Using the similar proof with Lemma 14, if we choose satisfy that
where is the intersection point of the straight line and the following characteristic
then, the blowup point satisfies that and .Until now, we have got solutions in the domains , , , and and the solutions can be defined
Step 4. Consider the Goursat problem in , , , and . In , we prescribe data as follows:
For , a characteristic passing by can be defined by
where is the intersection point of this characteristic and the characteristic . Along the characteristic , one has ; that is, . Clearly, the norm of is bounded. In order to estimate , we need to discuss . From (62), one has . Then, . According to Assumption 2, there at least exists a point such that . Then, = . Note that , there exists such that . As a result, we need to choose satisfy that
where . With (63), the Goursat problem has a unique global solution in .Similarly, in , we prescribe data as follows:
For , a characteristic passing by can be defined by
where is the intersection point of this characteristic and the characteristic . Along the characteristic , one has , that is, . Clearly, the norm of is bounded. In order to estimate , we need to discuss . From (65), one has . Then, . According to Remark 6, there at least exists a point such that . Then, = . Note that , there exists such that . As a result, we need to choose satisfy that
where . With (66), the Goursat problem has a unique global solution in .In addition, with the same analysis, the Goursat problem always has a unique global solution in , . Therefore, under the conditions (63) and (66), we have got solutions in the domains , , , and the solutions can be defined
We have constructed solutions in the domain . Let , and ; then there exist such that systems (10) (12) have a solution on the domain satisfying , , , , for . The proof is completed.

Remark 17. In [10], an optimal control model of distributed parameter systems (DPSs) is presented to discuss the polymer injection strategies. Compared with [10], the differences of our paper are (1) the considered model is a hyperbolic system and the maximum principle does not hold here; (2) the desired outputs can be achieved by controlling the boundary inputs.

4. An Example

In this section, an example is presented to demonstrate the effectiveness of our results.

Example 1. For system (10), we define that
where is a constant, which denotes the equilibrium of the initial and terminal states. , , and , , , are all constants. is also a constant, which will be chosen in the following.As a result, one has with and . Let , , and depends on and is also decided later. In the following, we will choose appropriate to satisfy conditions (57), (58), (63), and (66).According to Theorem 12, one has = = . Note that , an appropriate can be chosen to satisfy condition (58). Applying the same method, it is also easy to choose an appropriate to satisfy condition (57).Let be the intersection point of the characteristic (65) and the straight line . Because the straight line is constructed, we can choose that . As a result, (66) will be satisfied if condition (57) is satisfied.Similarly, let be the intersection point of the characteristic (62) and the straight line . Because the straight line is constructed, we can choose that . As a result, (63) will be satisfied if condition (58) is satisfied. So, there exists a positive constant to satisfy conditions (57), (58), (63), and (66). In addition, for the chosen , it is easy to find out a constant to satisfy and = . Therefore, all the conditions in Theorem 12 can be satisfied by using the constructed method. That is, the conclusion of Theorem 12 is valid.

Remark 18. Compared with [20, 21], the difference in this paper is the fact that one cannot avoid the phenomenon of blowup. Hence, one important goal for EORs is to find the controllers and to make sure that the blowup points of EORs are beyond the domain we fixed. As a result, there exist more complications in the controlling process as described in Section 3. The example we give here is to illustrate that the controllers are achievable and it can be applied in real problems.

5. Conclusions

In this paper, we have discussed the exact boundary controllability of a class of enhanced oil recovery models. By using a constructed method, it has been shown that the enhanced oil recovery systems with nonlinear boundary conditions is exactly boundary controllable. Moreover, an interval of the control time has also been presented to be optimal. Finally, an example has been provided to illustrate the effectiveness of the obtained criterion.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was jointly supported by the National Natural Science Foundation of China under Grant no. 61203146, the Educational Commission of Sichuan province under Grant no. 12ZA198, the China Postdoctoral Fund under Grant no. 2013M541589, the Jiangsu Postdoctoral Fund under Grant no. 1301025B, the Open Fund of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation under Grant no. PLN1130, and the University Research Funds under Grants nos. 2012XJZ029, 2012XJZT005, and 2013XJZT004.